Card Range To Study

7 Cards in this Set

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Back

Limit

Let f:D  R be a function with D in R, xo in R is an accupoint of D. Moreover, let L be in R. We say f(s) has a limit L at xo iff the following holds:
-For every positive real number E, there exists a positive real d, such that whenever x is in D\{xo}, with |x-xo|<d, we have |f(x)-L|<E.

Sub-sequence

Let (an)(n in N) be a sequence and (nk)(k in N) be a sequence of positive integers such that n1<n2<n3<…<nk. The sequence (nk) is called a subsequence of (an).
In other words: deleting some or none of the terms from a sequence and renaming the remaining terms, with remaining terms in the same order as in the sequence. There are still infinitely many terms after some are deleted.

Increasing

sequence (an)(n in N) is increasing iff an<=an+1 for all positive integers

decreasing

Decreasing: a sequence is decreasing iff an>=an+1 for all positive integers n.

Monotone

a sequence is either increasing or decreasing

Summary of proofs with lim arithmetic

Take arbitrary sequence that converges to x0
Use thm 16lim(fxn)=A
Use some other rule for lim seq
Use thm 16 in other direction

Summary of proofs with convergent arithmetic

Set up convergent definition of an and bn with N1 and N2
(For multiplication add |an|<=M bounded)
(For division |bn|>c by lemma)
Define E
1. For addition-E=E/2
2. For multiple-E’=E/(/(|B|+M)>0
3. For division-E=|B|E*c

N=max(N1,N2) and then put everything together
Will use triangle inequality

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